Bias and error in multiple-choice tests

A formula for estimating real scores on a multiple-choice test from a knowledge of raw scores is derived. This formula does not involve the assumption of a binomial distribution of real scores as does the Calandra formula. Other important formulas derived show: the variance of real scores in terms of the variance of raw scores and the correlation between real scores and raw scores. If the variance of real scores (or of raw scores also) is binomial, the regression of real scores on raw scores is linear; but, otherwise the regression is curvilinear. Yet the linear estimating formula is a close approximation to the curvilinear relationship. Factors affecting the regression of real scores on raw scores and the correlation coefficient are: (1) the number of choices per question; (2) the number of questions answered; (3) the ratio of the average group raw score to the variance of raw scores.